Research Projects of the Algebraic Geometry Group

The research group has widespread research interests in algebraic geometry and its interaction with theoretical physics.

Several members of the group work on moduli spaces. One area concerns moduli spaces of algebraic varieties with trivial canonical bundle: abelian varieties, K3 surfaces and irreducible holomorphic symplectic manifolds, where we also explore the relationship with modular forms. Our interest lies in understanding the geometry of these moduli spaces as well as their Chow groups and their topology. We also investigate moduli spaces of curves, including spin curves and Prym curves. (Hulek et al.)

In several projects the group explores the connections of geometry and arithmetic. The main objects of study in this context are algebraic surfaces, Calabi-Yau threefolds and irreducible holomorphic symplectic manifolds. Notably, we investigate K3 surfaces with high Picard number or automorphisms. There are close connections with modular forms, class group theory, and Shimura varieties. In this context elliptic fibrations play an important role. For Calabi-Yau threefolds we have achieved several results on modularity. Currently our focus lies on the development of new constructions and their connections to string theory. (Schütt et al.)

A further main area of interest of our research group is singularity theory. Here we investigate singular points of real and complex algebraic varieties. This leads to many questions related to algebraic geometry and differential topology. Special topics are singularities of complex hypersurfaces and their monodromy as well as indices of vector fields and 1-forms on singular varieties. We also investigate the relationship to string theory. A special point of interest is the strange duality between weighted homogeneous polynomials which corresponds to the mirror symmetry of Landau-Ginzburg models. (Ebeling et al.)

Algorithmic aspects in algebraic geometry and singularity theory are a further focus of the group. These considerations not only contribute to a better theoretical understanding, but also to implementable algorithms which in turn provide useful tools for a more experimental approach. The topics explored here range from desingularization and its applications to geometric questions arising from the study of polynomial systems in neighbouring fields like e.g. group theory. (Frühbis-Krüger et al.)

Another scope of the group's research is the relationship between a complex manifold and its Hodge structure, the so-called Torelli problem. This is very well understood for curves and for varieties with trivial canonical bundle, which in particular allows to describe their moduli spaces. However, there is still a lot to explore in the context of varieties of general type. Understanding the variations of the Hodge structure of such manifolds, as well as applications to the study of their moduli spaces, is the guiding line of the Riemann Junior Research Group. (González-Alonso et al.)